Nilpotence and Stable Homotopy Theory II

Nilpotence And Stable Homotopy Theory Ii-Free PDF

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Nilpotence II 2,5 Endomorphisms up to nilpotents 37. 5 1 N endomorphisms 37,5 2 Classification of N endomorphisms 39. 5 3 Some technical tools 41,5 4 N endomorphisms and thick subcategories 43. 5 5 A spectrum with few nonnilpotent self maps 46,5 6 Proof of Theorem 5 4 48. A Proof of Theorem 4 12 49,Introduction, This paper is a continuation of 7 Since so much time has lapsed.
since its publication a recasting of the context is probably in order. In 15 Ravenel described a series of conjectures getting at the. structure of stable homotopy theory in the large The theory was. organized around a family of higher periodicities generalizing. Bott periodicity and depended on being able to determine the. nilpotent and non nilpotent maps in the category of spectra There. are three senses in which a map of spectra can be nilpotent. Definition 1,i A map of spectra,is smash nilpotent if for n 0 the map. f n F n X n,ii A self map,is nilpotent if for n 0 the map. f n kn F F,Nilpotence II 3, from the sphere spectrum to a ring spectrum is nilpotent if it is. nilpotent when regarded as an element of the ring R. The main result of 7 is, Theorem 2 In each of the above situations the map f is nilpo. tent if the spectrum F is finite and if M U f 0, In case the range of f is p local the condition M U f 0 can be.
replaced with the condition BP f 0, The purpose of this paper is to refine this criterion and to pro. duce some interesting non nilpotent maps Many of the results of. this paper were conjectured by Ravenel in 15, Let K n be the nth Morava K theory at the prime p see 1. i Let R be a p local ring spectrum An element R is. nilpotent if and only if for all 0 n K n is nilpotent. ii A self map f k F F of the p localization of a finite. spectrum is nilpotent if and only if K n f is nilpotent for all. iii A map f F X from a finite spectrum to a p local spectrum. is smash nilpotent if and only if K n f 0 for all 0 n. Of course the hypothesis p local can be dropped if the condition. on the Morava K theory is checked at all primes, At first the criterion of this theorem seems less useful than the. one provided by 7 Using Theorem 3 to decide whether a map. is nilpotent or not requires infinitely many computations On the. other hand Morava K theories are often easier to use than complex. cobordism Theorem 3 also determines which cohomology theories. detect the nonnilpotent maps in the category of spectra. Definition 4 A ring spectrum E is said to detect nilpotence if. equivalently, i for any ring spectrum R the kernel of the Hurewicz homomor. phism E R E R consists of nilpotent elements,Nilpotence II 4.
ii a map f F X from a finite spectrum F to any spectrum X. is smash nilpotent if 1E f E F E X is null homotopic. Corollary 5 A ring spectrum E detects nilpotence if and only if. for all 0 n and for all primes p, Now let C0 be the homotopy category of p local finite spectra. and let Cn C0 be the full subcategory of K n 1 acyclics The. Cn fit into a sequence,Cn 1 Cn C0, This is a nontrivial fact That there are inclusions Cn 1 Cn is. essentially the Invariant Prime Ideal Theorem See 15 That the. inclusions are proper is a result of Steve Mitchell 12. Definition 6 A full subcategory C of the category of spectra is. said to be thick if it is closed under weak equivalences cofibrations. and retracts ie, i An object weakly equivalent to an object of C is in C. ii If X Y Z is a cofibration and two of X Y Z are in C. then so is the third,iii A retract of an object of C is in C. Theorem 7 If C C0 is a thick subcategory then C Cn for. Theorem 7 is in fact equivalent to the Nilpotence Theorem the. proof is sketched at the end of Section 4 It is often used in the. following manner, Call a property of p local finite spectra generic if the full sub.
category of C0 consisting of the objects with P is closed under cofi. brations and retracts To show that X Cn has a generic property. Nilpotence II 5, P it suffices by Theorem 7 to show that any object of Cn Cn 1. has P The proofs of the next few results use this technique. Theorem 3 limits the nonnilpotent maps in C0 they must be. detected by some Morava K theory The simplest type is a vn. Definition 8 Let X be a p local finite spectrum and n 0 A. self map v k X X is said to be a vn self map if,multiplication by a if m n 0. rational number,K m v is if m n 6 0,an isomorphism. nilpotent if m 6 n, It turns out that the property of admitting a vn self map is. Theorem 9 A p local finite spectrum X admits a vn self map if. and only if X Cn If X admits a vn self map then for N 0. X admits a vn self map,satisfying,vnp if m n,0 otherwise.
The vn self maps turn out to be distinguished by another prop. Definition 10 A ring homomorphism,is an F isomorphism if. i the kernel of f consists of nilpotent elements and. Nilpotence II 6,ii given b B bp is in the image of f for some n. Two rings A and B are F isomorphic A F B if there is an. F isomorphism between them,Theorem 11 Let X Cn Cn 1 The K n Hurewicz homomor. phism gives rise to an F isomorphism,11 1 center X X F. Fp vn n 6 0, The description of spectra as cell complexes encourages the in.
tuition that the endomorphism rings of finite spectra approximate. matrix algebras over the ring S 0 This would suggest that the. centers of these rings are generated by the maps obtained by smash. ing the identity map with a map between spheres an impossibil. ity by Theorem 11 A more accurate description might be that. the Morita equivalence classes of these rings are determined by. the integer n of Theorem 11 This integer invariant can also be. thought of as determining the birational equivalence classes of. finite spectra For more on this analogy see 9, There is a less metaphorical interpretation of the integer which. occurs in Theorems 9 and 11, Definition 12 Let X be a spectrum The Bousfield class of X. denoted hXi is the collection of spectra Z for which X Z is not. contractible, The Bousfield classes of spectra are naturally ordered by inclu. sion though the relation is indicated with rather than. For a finite spectrum X let Cl X N P denote the set. of pairs n p for which K n X 6 0 at p Here N is the set of. nonnegative integers and P is the set of primes, Theorem 13 Let X and Y be finite spectra Then hXi hY i if. and only if Cl X Cl Y, Theorem 13 affirms Ravenel s Class Invariance conjecture 15.
Nilpotence II 7, Proof of Theorem 13 Since hXi hY i if and only if hX p i. hY p i for all primes p we may localize everything at a prime p. For a fixed Y the property of X,is a generic property It follows that the class. X hXi hY i, is equal to Cm for some m Suppose that Y Cn Cn 1 We need. to show that m n Since hY i hY i m n But if X,hXi 6 hY i since K n 1. hXi so m 6 n,Acknowledgements, Most of the results of this paper date from 1985 and there have.
been many people who helped shaped the course of the results. Special thanks are due to Emmanuel Dror Farjoun whose prodding. eventually led to the formulation of Theorem 7 and to Clarence. Wilkerson for helpful conversations concerning the proof of Theo. rem 4 12 Even deeper debts are owed to Doug Ravenel for formu. lating such a beautiful body of conjectures and to Mark Mahowald. for placing in the hands of the authors the tools for proving results. like these Finally the first author would like to dedicate his con. tributions to this paper to Ruth Randi and Rose,Notation and conventions. For the most part we will work in the homotopy category of spec. tra Of course to form things like the induced map of cofibers. requires choosing a diagram in a model for the category of spectra. introducing a certain ambiguity into the resulting map This am. biguity plays only a very small role in this paper and is dealt with. each time it comes up, The cofiber of a map X Y will be written with the cone. coordinate on the right,Y f CX Y X I,Nilpotence II 8. With this convention the cofiber of, is Z Y f CX modulo associativity of the smash product. not just isomorphic to it With this convention the cofiber of. Y Y f CX is X S 1 which is isomorphic but not equal to. X This avoids encountering the troublesome sign that can crop. up when trying to relate the connecting homomorphism in a cofi. bration with the connecting homomorphism in some suspension of. the cofibration, The assumption that a spectrum is finite is made several times.
In contexts when the the category in mind is the category of p local. spectra this term is used to refer to a spectrum which is weakly. equivalent to the p localization of a finite spectrum The only prop. erty of finite spectra that is used is that the set of homotopy classes. of maps from a finite spectrum to a directed colimit is the colimit. of the maps, In general and object of a category with this property is said to be. small It can be shown that the small objects of the category of p. local spectra are precisely the objects which are weakly equivalent. to the p localizations of a finite spectrum,A spectrum X is connective if k X 0 for k 0 It is. connected if k X 0 for k 0 Thus connected and 1, connected are synonymous Similarly a graded abelian is con. nective if the homogeneous part of degree k is zero for k 0 A. graded abelian group is connected if the homogeneous component. of degree k is zero for k 0, The Eilenberg MacLane spectrum with coefficients in an abe. lian group A will be denoted HA To be consistent with this the. homology of a spectrum X with coefficients in A will be denoted. Finally the suspension of a map will always be labeled with the. same symbol as the map,1 Morava K theories,Nilpotence II 9.
1 1 Construction, The study of a ring is often simplified by passage to its quotients. and localizations The same is true of ring spectra though con. structing quotients and localizations can be difficult In good cases. the following constructions can be made, Suppose that E is a ring spectrum and that E R is commu. tative Given x R define the spectrum E x by the cofibration. If x is a non zero divisor then E x is isomorphic to the ring. R x In good cases E x will still be a ring spectrum and the. will be a map of ring spectra Given a regular sequence. one can hope to iterate the above construction and form a ring. spectrum E x1 xn with,E x1 xn R x1 xn,and such that the natural map. is a map of ring spectra,Localizations, Let E and R be as above and suppose that S R is a multiplica. tively closed subset Since S 1 R is a flat R module the functor. is a homology theory S 1 E In good cases it is represented by a. ring spectrum and the localization map by a map of ring spectra. Nilpotence II 10,1 2 Spectra related to BP, When the ring spectrum in question is BP the above constructions.
can always be made using the Baas Sullivan theory of bordism. with singularities See 4 13 17 for the details,Recall that BP Z p v1 vn with vn 2pn 2 To. fix notation take the set vn to be the Hazewinkle generators 8. For 0 n the ring spectra K n and P n are defined by the. isomorphisms,K n Fp vn vn 1,P n Z p vn vn 1, with the understanding that they are constructed from BP using. a combination of the above methods It is also useful to set. There are maps P n P n 1 and the limit,is the Eilenberg MacLane spectrum HFp. Proposition 1 1 The Bousfield classes of K n and P n are re. hP n i hK n i hP n 1 i,Consequently,hBP i hK 0 i hK n i hP n 1 i. Proof The proposition follows from the next two results of. Ravenel 15, Proposition 1 2 Let v k X X be a self map of a spectrum.
X Let X vX and v 1 X denote the cofiber of v and the colimit of. the sequence,Nilpotence II 11, respectively Then there is an equality of Bousfield classes. hXi hX vXi hv 1 Xi, Proposition 1 3 There is an equality of Bousfield classes. hvn 1 P n i hK n i,1 3 Fields in the category of spectra. The coefficient ring K n is a graded field in the sense that all of. its graded modules are free This begets a host of special properties. of the Morava K theories, Proposition 1 4 For any spectrum X K n X has the homo. topy type of a wedge of suspensions of K n, Proof Choose a basis ei i I of the free K n module K n X.
and represent it as a map,S ei K n X,The composition. K n S ei K n K n X K n X,is then a weak equivalence. Proposition 1 5 For any two spectra X and Y the natural map. 1 5 1 K n X K n K n Y K n X Y,is an equivalence,Nilpotence II 12. Proof Consider the map 1 5 1 as a transformation of func. tors of Y The left side satisfies the Eilenberg Steenrod axioms. since K n Y is a flat in fact free K n module The right side. satisfies the Eilenberg Steenrod axioms by definition The trans. formation is an isomorphism when Y is the sphere hence for all. Proof of Corollary 5 If for some n K n E 0 then E does. not detect the nonnilpotent map,If K n E 6 0 then by Proposition 1 4. so the result reduces to Theorem 3, Propositions 1 4 and 1 5 portray the Morava K theories as being.
a lot like fields One formulation of Theorem 3 is that they are the. prime fields of the category of spectra, A skew field is a ring all of whose modules are free. Definition 1 6 A ring spectrum E is a field if E X is a free E. module for all spectra X,This property also admits a geometric expression. structure of stable homotopy theory in the large The theory was organized around a family of higher periodicities generalizing Bott periodicity and depended on being able to determine the nilpotent and non nilpotent maps in the category of spectra There are three senses in which a map of spectra can be nilpotent De nition 1 i A map of

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